1. Purpose of multipart@trinitas.mju.ac.kr     [ 한글 설명 ] 
Computation of the values of the (graded) partition function p_M(j) over a multiset M for various numbers j <= n, and immediate return to the e-mail sender of the computed results.

2. Introduction
We first recall that {1,2,2,3} isa multiset since the number 2 appears more than once.  Given a multiset M and a natural number n, the total number of ways of writing n as a non-increasing sum of members of M is denoted by p_M(n). Recently Prof. Sun T. Soh discovered a new recursive formula for p_M(n) that is a generalization of the Euler's method of direct expansion of its generating function: 

Soh's (quasi-) recursive formula for p_M(n)

We use Soh's formula in this section of our InetCompu. The time efficiency of his recursive formula is O(n^2), and by choosing a value for effciency control parameter f > 1 properly its global performance is further improved. 

3. Contributor: The program for p_M(n) we are using was developed in Reduce commands by Prof. Sun T. Soh, Dept of Math, Myong Ji Univ., Rep. of Korea. 

4. Note
For a trouble-free handling of your e-mail, we strongly recommend you to use MicroSoft Outlook Express, New mail > Alt+O > Alt+X (with No Encryption),  to send out an e-mail to multipart@trinitas.mju.ac.kr   .   

5. How to do: Send an e-mail with plain text style (for instance, in the case of MicroSoft Outlook Express, New mail > Alt+o > Alt+x (with No Encryption)) to multipart@trinitas.mju.ac.kr whose main body should consist of, for example,

     input: 
n:=1000$
multiset:={1,2,2,3,3,3,4,5,5,5,5,6,7,8,9,9,10}$
f:=2$
     end input:  

where 
(i) the third line, multiset:={1,2,2,3,3,3,4,5,5,5,5,6,7,8,9,9,10}$ , can be replaced with, multiset:={{1,1},{2,2},{3,3},{4,1},{5,4},{6,1},{7,1},{8,1},{9,2},{10,1}}$ ,  
(ii) a different number other than 2 may be chosen for efficiency control parameter f  >1 by the sender, and 
(iii) if you don't write the third line, multiset:={1,2,2,3,3,3,4,5,5,5,5,6,7,8,9,9,10}$  in between,  input:   and   end input: , then it will assume that your multiset is, multiset:={1,2,...,n}$ . 

Note: For a complicated or large multiset, a different value other than 2 may be chosen for f > 1 using the formula: 

f:=[exp(sqrt( ln(2)*ln(N) ))]$ 

where [m] means the largest number <= m  and N is the cardinality of the multiset under consideration. (It is usually most efficient when f:=2$, unless multiset is quite large). 

Upon the arrival of the e-mail, the values of p_M(j)'s for various j's from 1 to 1000 are automatically computed and sent back to the e-mail sender immediately.

[Reminder] When the requested job is computationally not very complicated, it should be quite the case that you will receive the result within a few Minutes.  But, our response consisting of the computed results can not be delivered to the sender properly, if either there is a spelling mistake in sender's e-mail address or sender's mail box is already filled with too many of other e-mails. Thus, if you do not receive the results although you have waited for some time, then please check your mail account to correct the above trouble-causing problems.  After that, try again according to the procedure described in Section 5 above.